Bordered Floer homology is a variant of Heegaard Floer homology adapted to manifolds with boundary. For a large class of three-manifolds with torus boundary, I will present a geometric interpretation of these invariants in terms of immersed curves on the boundary torus. In this setting, the pairing theorem in bordered Floer homology can be reinterpreted in terms of intersection between curves. As one application, we can use this new interpretation of bordered Floer homology to recover and extend a recent result concerning toroidal L-spaces. This is joint work with Jake Rasmussen and Liam Watson.